The article "The Shape of Math To Come" by Alex Kontorovich explores the interaction between computational tools and mathematical practice, reflecting on the potential impacts of artificial intelligence and formal verification systems on research mathematics in the near future. It aims to provide insights into how these advancements may shape the field of mathematics.

The article presents a collection of useful mathematical functions and their plots that can be applied in graphics for filtering or smoothing signals. Each function is defined for input values within the range of [-1, 1] and includes variations based on different parameter values. The emphasis is on providing visual representations to aid understanding of these functions.

The article explores interesting numerical relationships in various bases, particularly focusing on the ratios of descending and ascending digit concatenations in those bases. It demonstrates that these ratios are nearly integers, specifically b - 2 for base b, and discusses the implications of floating-point precision in these calculations. The author also highlights the complementary nature of computational demonstrations and mathematical proofs.

The article discusses the spacing of circles on the Smith chart, explaining how to achieve desired circle distributions by working backward from the chart to the z plane. It details the differences between complete and incomplete circles and the methods to space them uniformly using the inverse function. The author notes that typical Smith charts aim for a compromise between even spacing in both the w and z planes.

The document appears to be a PDF file related to the Braid Group, a mathematical concept in the field of algebra and topology. However, the content is not directly readable or accessible due to the PDF format and possible encoding issues. Further analysis or conversion would be necessary to extract specific information or insights from this document.

The article explores the concept of wordless thinking, particularly in the context of mathematicians who report solving complex problems without verbal or visual representations. It discusses how subconscious processing can lead to insights and the distinction between this type of thinking and more traditional, verbal thought processes. The author reflects on personal experiences with writing and the challenges of articulating thoughts clearly.

The article discusses the discovery of a shape called the Noperthedron, which is the first known convex polyhedron that cannot pass through itself, a problem that has puzzled mathematicians for centuries. This shape, proven by Jakob Steininger and Sergey Yurkevich, defies the previously held conjecture that all convex polyhedra have the "Rupert property," which allows one shape to pass through another of the same kind. The solution involved advanced theoretical work and extensive computational calculations.

The article discusses the importance of formalizing mathematics through the use of proof assistants like Lean, highlighting benefits beyond just error detection. It draws a parallel to TypeScript, emphasizing how formalization can enhance mathematical tools, facilitate analysis of mathematical trends, and improve collaboration through better documentation and version control. The author expresses a personal passion for engaging with mathematics and computer science through this formalization process.